Problem: $ A = \left[\begin{array}{rrr}5 & 0 & 4 \\ -1 & 1 & 4\end{array}\right]$ $ B = \left[\begin{array}{rr}-1 & 0 \\ 4 & 1 \\ 2 & 4\end{array}\right]$ What is $ A B$ ?
Answer: Because $ A$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A B = \left[\begin{array}{rrr}{5} & {0} & {4} \\ {-1} & {1} & {4}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{0} \\ {4} & \color{#DF0030}{1} \\ {2} & \color{#DF0030}{4}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{5}\cdot{-1}+{0}\cdot{4}+{4}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{-1}+{0}\cdot{4}+{4}\cdot{2} & ? \\ {-1}\cdot{-1}+{1}\cdot{4}+{4}\cdot{2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{-1}+{0}\cdot{4}+{4}\cdot{2} & {5}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{4} \\ {-1}\cdot{-1}+{1}\cdot{4}+{4}\cdot{2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{5}\cdot{-1}+{0}\cdot{4}+{4}\cdot{2} & {5}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{4} \\ {-1}\cdot{-1}+{1}\cdot{4}+{4}\cdot{2} & {-1}\cdot\color{#DF0030}{0}+{1}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}3 & 16 \\ 13 & 17\end{array}\right] $